NQS2011: Novel Quantum States in Condensed Matter: Correlation, Frustration, and Topology Yukawa Institute, Kyoto University. Kyoto, Japan 2011-11-24
Geometric field theory (“quantum geometry”) description of the fractional quantum Hall effect F. Duncan M. Haldane Princeton University • Now revealed: the missing ingredient in our understanding of incompressible quantum fluids that exhibit the FQHE: geometry of “flux attachment” • New topological quantum numbers of the incompressible state: “guiding center spins”
arXiv: 1106.3365, Phys. Rev Lett. 107.116801
Thursday, November 24, 11 Before we start, here is the “bottom line”
Action in terms of Chern Simons gauge fields plus metric(s) and their curvature gauge fields S = S dt H 0 • topological action Z ~ eAµ S = S ( a )+ ✏µ⌫ d3x ti + si ⌦ (g ) @ a 0 CS { iµ} 2⇡ ↵ µ ↵ ⌫ i ~ ↵ ! Z X spin connection of metric • Hamiltonian 2 u(g) 1 2 2 H = d r + d r d r0V (r r0) ⇢(r) ⇢(r0) 2⇡`2 2 Z B Z Z coulomb energy of geometry-dependent charge fluctuations correlation energy e ⇢ = ⇤ s↵K(g ) 2⇡ ↵ ↵ X
Gaussian curvature of metric
Thursday, November 24, 11 Landau quantization • “Dynamical momentum” “Landau orbit (inverse)metric” ⇧ p eA(r) det g =1 ⌘ (1) 1 ab 1 H = g ⇧a⇧b = ~!c(a†a + ) 2m 2 • shape of Landau orbit defines a “unimodular” (determinant 1) positive definite 2 x 2 “Landau orbit” spatial metric tensor gab • complex-plane Schrödinger representation of Landau level raising and lowering operators:
z @ z⇤ @ [a, a†]=1 a† = a = + 2 @z⇤ 2 @z (harmonic oscillator)
Thursday, November 24, 11 • The Landau-level raising and lowering operators commute with a second set of Harmonic oscillator operators, which change the position of the “guiding centers” of the Landau orbits: z @ z @ a = ⇤ + a¯ = + 2 @z 2 @z⇤ z @ z⇤ @ a† = a¯† = z z⇤ 2 @z⇤ 2 @z $ [a, a†]=1 [¯a, a¯†]=1
[a, a¯†]=0 [a, a¯]=0
[a†, a¯†]=0 [a†, a¯]=0
Thursday, November 24, 11 • W algebra of Generators of “area preserving diffeomorphisms”1 of the Landau level” n n Wmn =(¯a†) (¯a)
(1) 1 (1) H = ~!c(a†a + 2 ) [Wmn,H ]=0
Thursday, November 24, 11 • Lowest Landau level states : @ z⇤ a =0 + (z,z⇤)=0 | i @z 2 Heisenberg form Schrödinger form ⇣ ⌘ holomorphic 1 z⇤z (z,z⇤)=f(z)e 2 • filled lowest Landau level many-electron state : 1 a 0 =0 a¯ 0 =0 z⇤zi i i (z z ) e 2 i | i | i i j = a¯† a¯† 0 i Thursday, November 24, 11 Laughlin state Schrödinger form 1 ⌫=1/m m z⇤z ( z ,z⇤ )= (z z ) e 2 i i L { i i } i j i Thursday, November 24, 11 • The Laughlin state contains all the essential ingredients of the FQHE • It can be interpreted in terms of “flux attachment” (Chern-Simons theories) • It naturally leads to fractional charge and statistics quasiparticles • It is exhibits topological order and long- range topological entanglement • The original Schrödinger (not Heisenberg) form can be very misleading, and it “hides” the fundamental quantum geometry of the FQHE Thursday, November 24, 11 • overview of FQHE Thursday, November 24, 11 0.9 laughlin 10/30 0.8 For at least 20 years, ΔE • 0.7 goes into most “theoretical” 0.6 continuum (2 quasiparticle work (as opposed to 0.5 + 2 quasiholes) numerical simulation) 0.4 TQFT has been “topological” 0.3 0.2 “roton” in character. 1 TQFT 0.1 Laughlin ⌫ = 3 0 • Finite-size exact 0 0.5 1 1.5 2 2.5 3 3.5 diagonalization (up to gap k B 20 particles) confirms incompressibility that microscopic Hamiltonians exhibit • Energy-scale analysis shows incompressibility the gap must be of order e2 ~ E ` = p ⇠ 4⇡✏0✏`B B eB | | Thursday, November 24, 11 Topological Quantum Field Theory • (Abelian) Chern-Simons theory ~ e S = d3x KIJ✏µ⌫ a @ a + tI A @ a 4⇡ Iµ µ J 2⇡ µ ⌫ Z • Choose gauge aI0 = 0 ~ e S = d3x 1 ✏µ⌫ A @ A + KIJ✏aba @ a + tI ✏abA @ a 2 H µ ⌫ 4⇡ Ia t jb 2⇡ a t Ib Z 2 e I 1 J linear in time-derivative H = (t K t ) 2⇡~ IJ KIJ is an integer topological matrix, tI is an integer vector Topological degeneracy on a • det K g 2-manifold of genus g | | Thursday, November 24, 11 Topological quantum field theory • The TQFT description of FQHE assumes the existence of incompressibility. • It can classify the different types of topological excitations, and what happens when external agents move them around braided paths or and selection rules for fusing them. • TQFT models the systems with a Lagrangian that is linear in time-derivatives. • The weak point of TQFT follows from this: The Hamiltonian has the simple form: H =0 Thursday, November 24, 11 • TQFT is a fundamentally incomplete description of the FQHE It does not even know about the • 2 fundamental magnetic area 2⇡`B =( 0/B) • It does not describe the relative energies of the point-like topological excitations it classifies, just their electric charge and mutual statistics. Thursday, November 24, 11 Origin of incompressibility? Various “narratives” have been developed • Ginzburg-Landau Chern-Simons theory suggests that it could be explained by a theory where CS flux is “attached” to Galilean-invariant non-relativistic particles with mass m. • The composite fermion (CF) idea (Jain) says it arises because composite fermions fill “effective Landau levels”. “Effective Hamiltonian Theory” (Shankar and Murthy) tries to implement this with an uncontrolled Ansatz. CF method produces model wavefunctions for 2/5, 2/7, .. . series of FQHE states that can be used as successful variational states with numerically-evaluated energies, but provides no analytic insight. Thursday, November 24, 11 Are these “narratives” just “conforting make-believe stories” that reassure us we have some understanding, even if it is non- quantitative? • These approaches have not been based on microscopic analysis, and attempt to solve the problem by appealing to superficially- plausible analogies • Dimensional analysis suggests gaps must be related to e2 1 = 4⇡✏ ✏ ` ✓ 0 ◆ B We may hope for something more quantitative Thursday, November 24, 11 • The clue to a quantitative approach is provided by seminal work of Girvin MacDonald and Platzman (1985) • It was never properly interpreted and followed up, but turns out to be the only source of correct physics on which a microscopic picture can be based • The new results presented here can be viewed as a translation of GMP into a “geometric field theory” Thursday, November 24, 11 d2q`2 GMP: v = B v(q) k k 2⇡ | | Is finite Z N 2 2 d q`B iq Ri H = v(q)⇢(q)⇢( q) ⇢(q)= e · (needs 2⇡ Z i=1 regularization) Hamiltonian X [Ra,Rb]= i`2 i j B ij 1 2 [⇢(q), ⇢(q0)] = 2i sin( q q0` )⇢(q + q0) 2 ⇥ B Fundamental Lie algebra Landau level filling 2 ⇢(q)=⇢(q) 2⇡⌫ (q`B) also obeys this algebra • The one step missed by GMP (regularization) lim ⇢( q)=0 u( ) 0 replace ⇢(q)by ⇢(q) 1 =su( ) ! u(1) 1 Thursday, November 24, 11 regularized generator of translations • ~ 1 b Pa = `2 lim 0 ✏ab q ⇢( q) B ! r 2 Components commute ~ [Pa,Pb]=i✏ab 2 ⇢(0)=0 `B after regularization! (this solves a fundamental problem in taking the thermodynamic limit of the FQHE) Translationally-invariant vacuum state P 0 =0 0 ⇢(q) 0 =0 a| i h | | i 2 0 ⇢(q) ⇢(q0) 0 =2⇡S(q) (q` + q0` ) h | | i B B Guiding-center structure function: the importance of this correlation function is central to the theory of FQHE incompressibility Thursday, November 24, 11 • correlation energy per magnetic area 2 E0 d q N = v(q)S(q) = ⌫ N 2⇡ N • S(q) is theZ ground-state structure factor, which describes the zero-point fluctuations of the Ri • make an area-preserving shear deformation of the ground state, i.e., of S(q); the energy increase will be quadratic in the deformation b cb qa a + ac✏ qb E 1 abcd ! = G ab cd N 2 symmetric guiding-center shear modulus Thursday, November 24, 11 • Motivated by Feynman’s theory of the roton in 4He, GMP used the single-mode approximation as a variational ansatz for the collective excitation: A(q) sma(q) = ⇢(q) exact "(q)=E(q) Eexact = | i | 0 i 0 S(q) 2 2 1 d q0`B 1 2 2 A(q)= v(q0)(S(q0 + q)+S(q0 q) 2S(q0)) 2sin q q0` 2 2⇡ 2 ⇥ B Z • GMP did not interpret A(q), but evaluated it numerically, using the in fact Laughlin state S(q) obtained by Monte Carlo methods. Thursday, November 24, 11 • In the long wavelength limit, the GMP result can be written as • This turns out to be an equality for systems with a single collective mode (single-component FQHE states, as well as Wigner-lattice states with one electron per unit cell) • As GMP recognized, if the collective mode is gapped (i.e., the state is incompressible), S(q) must be quartic at long wavelengths. This was their fundamental insight into FQHE incompressibility. Thursday, November 24, 11 • From this we learn that the fundamental stiffness of the incompressible FQHE states is their resistance to area-preserving distortions that change the shape of the correlation hole around a guiding center from the shape that minimizes the energy. • The collective degrees of freedom can be described as one (or more) UNIMODULAR positive definite real-symmetric spatial metric tensor fields As a quadratic form, this describes a local “shape of a circle” Thursday, November 24, 11 • Physically, it is the shape of the “attached flux” of the “composite bosons” that condense in the Chern-Simons “flux attachment picture: area-preserving shape deformation of e e the exclusion region costs correlation energy region with 3 flux quanta surrounding the electron. at 1/3 filling, an electron with Other electrons are excluded from this region 3 “attached” flux quanta (analogy is a Hubbard model behaves like a neutral boson lattice site) Thursday, November 24, 11 • The metric has non-commuting elements that fluctuate around a value • Charge fluctuations associated with this metric are given by a “guiding-center spin” elementary fractional (2s = integer, topologically- quantized by Gauss-Bonnet in charge e/q incompressible states) Gaussian curvature (Brioschi formula) This is why S(q) ~ q4 Thursday, November 24, 11 • A 2+1-d space-time metric is given by • There is nothing that propagates on the Hall surface with speed c, and there is no Lorentz symmetry. Absolute simultaneity (unretarded Coulomb interaction) is allowed in the non-relativisic model. Thursday, November 24, 11 • Wen and Zee (1992) provided the infrastructure necessary to construct the extension to the topological Lagrangian when they described coupling to the extrinsically-derived curvature of an embedded surface on which the electrons move • In their case, the embedded surface in 3D Euclidean space and its normal are g = @ r~ @ r~ ab a · b Thursday, November 24, 11 • Wen and Zee extended (Abelian) CS theory: Coupling to “spin connection” curvature is a quantized spin curvature gauge field Gaussian curvature current (conserved) Thursday, November 24, 11 • Wen and Zee treated the metric induced on the surface by the flat 3D Euclidean metric of the space in which the surface was embedded, with a local SO(2) isotropy of rotations around the normal. • They thought the (2D orbital) “spin” was only meaningful if this rotational invariance was broken and the new features were only of formal interest for compactification of FQHE on a sphere (“shifts”, etc.), and would not survive disorder. This turns out to be false: “Spin” is quantized here by Gauss-Bonnet (topological), not dependent on rotational invariance Thursday, November 24, 11 • In the new interpretation, the metric(s) are tensor fields describing the shape fluctuations of the Composite bosons. • The physical (embedded) surface is flat. (atomically clean surfaces like graphene or epitaxially grown surfaces strongly resist Gaussian curvature of their physical shape). Thursday, November 24, 11 • Happily, Gaussian curvature formulas are intrinsic, and do not depend on the physical origin of the metric. One metric and its spin vector for each At last! independent “composite a Hamiltonian! boson” in the multicomponent case. Thursday, November 24, 11 • The Hamiltonian correlation energy + 1 d2r K K quadratic in local 2 ↵↵0 ↵ ↵0 curvatures Z X↵↵0 1 2 2 + 2 d r d r0V (r, r0) ⇢(r) ⇢(r0) Z Z long-range unretarded Coulomb Thursday, November 24, 11 • This seems to reproduce all the basic phenomenology of incompressibility: • For single CS field model (Laughlin-type), the Girvin-MacDonald-Platzman collective mode spectrum is reproduced at long wavelengths • The Hall viscosity and Q4 long wavelength behavior of the guiding-center structure factor S(Q) are reproduced • Effective theory will allow calculation of the energies of different types of topological excitations in terms of its parameters Thursday, November 24, 11 Back to “square one”: rexamine the description of the FQHE • electrons moving on a 2D surface with a magnetic flux density passing through it ab = p eA (r ) ⇥aAb(r)=B ia ia a i 1 ab 1 iq (r r ) H = g + V (q) e · i j 2m ia ib A i q i Landau Coulomb orbit + equipotentials Thursday, November 24, 11 • Most work on the fractional Hall effect implicitly assumes that the “circles” defined by the Coulomb point charge equipotentials and the Landau orbits are congruent • There is NO general reason for this to be true, one derives from the 3D dielectric tensor, the other from 2D bandstructure. My claim: the “simplification” of treating the Coulomb and Galileian metrics as identical (which gives the system rotational invariance) has hidden a fundamental geometric property of the FQHE Thursday, November 24, 11 decomposition into “guiding mab = mgab centers” and dynamical momenta a b b mab r r =const. a = pa eAa(r)=mabv e dynamical momentum 2D Landau x in a magnetic field R orbits “guiding center” b a a a a ab [ a,R ]=0 r = R + r eB r = ⇥ ⇤b • Non-commutative geometry of guiding centers! [Ra,Rb]= i⇥2 ab B Thursday, November 24, 11 High-field limit 2 ~ 1 2 ~ B V (q)f(q) ⌘ m⇥2 A B q Landau energy X form-factor of lowest Landau level b (depends on Galileian metric) [ a,R ]=0 f(q)=exp 1 gabq q 2 4 a b B “left-handed” ⇡ “right-handed”R act in Hilbert space L act in Hilbert space L R H R (g) 2 H 0 ~ a b ab 2 | ⇤⇥| ⇤ | ⇤ [⇥a, ⇥b]=i ab 2 [R ,R ]= i ⇥B B • in the high-field limit, the low-energy eigenstates of H become unentangled products of states in“left” and “right” Hilbert spaces Thursday, November 24, 11 L(g) R | 0 ⇥ | ⇥ • “Left variable” state is a trivial fully-symmetric coherent harmonic oscillator state that depends only on the Galilean metric a (g) L(g) =0,i=1,...,N i | 0 • “Right variable” state is a non-trivial eigenstate of N 1 2 iq Ri HR = V (q)f(q) q q q = e · 2A q i=1 X 1 2 X [ , ]=2i sin( q q0⇥ ) q q0 2 ⇥ B q+q0 Thursday, November 24, 11 Quantum geometry (Algebra , Hilbert space , Hamiltonian H ) (the “triple” inA Alain Connes definitionH of non-commutative geometry A (quantum geometry) • Discard the trivial “left variables”, and work only in the Hilbert space of the “right variables” (guiding centers). This makes numerical diagonalization tractable for finite N, NΦ . • Without the “left variables”, the notion of locality needed by classical geometry, and Schrödinger’s formulation of quantum mechanics is lost! Thursday, November 24, 11 • Wavefunctions are only possible after “left” and “right” states are “glued” back This is where together! the guiding center physics L R is! ↵( ri ,g)= ri 0 (g) ↵ wavefunction{ } h{ }| | i⌦| i ⇥ ⇤ Galileian basis of simultaneous eigenstates of Galileian metric the commuting set of metric operators ri Thursday, November 24, 11 • The wavefunction depends on the Galilean R metric gab in addition to |Ψ α〉 • This means it is not a “pure” representation of |ΨRα〉 because it involves extraneous elements! • The “pure” FQHE physics should only derive from guiding center physics N 1 2 iq Ri HR = V (q)f(q) q q q = e · 2A q i=1 X 1 2 X [ , ]=2i sin( q q0⇥ ) q q0 2 ⇥ B q+q0 (+ Hilbert space ) Thursday, November 24, 11 • But.... a great part of successful FQHE theory is based on wavefunctions! (e.g. Laughlin) 1 z⇤z ( r )=F ( z ) e 2 i i { i} { i} holomorphic i (g)ra Y z = a i i p2⇥ 1 B ai = zi + 2 zi⇤ The definition of zi depends on the Galileian metric a ( r )=0,j=1,...,N j { i} (lowest Landau level condition) Thursday, November 24, 11 Laughlin wavefunction F q ( z )= (z z )q L { i} i j i Thursday, November 24, 11 • I just argued that holomorphic wavefunctions were not “faithful” representations of quantum-geometric guiding- center states because they involve the Galileian metric • Another definition of Laughlin state follows from “Haldane pseudopotentials”. At filling 1/q, it is the only zero-energy eigenstate of: q 1 HR(g)= VmPm(g),Vm > 0 e e Text m=0 1 X 1 2 2 Pm(g) projects on pairs of 2 2 2 qg ⇥B Pm(g)= Lm(qg ⇥B)e q q 2N guiding centers with relative q X angular momentum m 2 ab Laguerre polynomial qg = g qaqb argument depends on a metric! (inverse) metric Thursday, November 24, 11 • This defines a family of Laughlin states parametrized by a unimodular “guiding center metric” g¯ab note the bar above g P (¯g) q (¯g) =0,m=0, 1,...,q 1. m | L ⇥ • The guiding center metric parametrizes an elliptic deformation of the shape of the “correlation hole” surrounding the electrons relative to the “circular” shape of the Landau orbits. • The guiding center metric is NOT fixed by any one- particle physics, but should be viewed as a true variational parameter that is chosen to minimize the correlation energy! Thursday, November 24, 11 1 2 HR = V (q)f(q) q q 2A q X depends on depends on Coulomb metric Galileian metric • If the Coulomb and Galileian metrics coincide, the correlation energy is lowest when the guiding-center metric (as a variational parameter) is equal to them both • If they are not equal, the energy is minimized by choosing the guiding center metric intermediate between them. Thursday, November 24, 11 • While TQFT can classify different “vortices” by electric charge and braiding statistics, it cannot say what their relative energies are, or how the process of fusion of two vortices proceeds. • Till now, there has been no viable analytically tractable effective theory that can “explain” the origin of incompressibility, and its properties. • Heuristic ideas include: • Analogy with superfluids (Ginzburg-Landau +Chern Simons) • Composite fermions fill “Landau levels” in analogy with the integer effect, where the Pauli principle explains incompressibility • Unfortunately, these are “narratives” rather than tractable effective theories. Thursday, November 24, 11 The Laughlin state: two different interpretations • The wavefunction: 1 q z z⇤ = (z z ) e 2 i i L i j i ai L =0 lowest-Landau-level condition Thursday, November 24, 11 1 q z z⇤ = (z z ) e 2 i i L i j i “The Laughlin state places q zeroes of the wavefunction as a function of any zi at the locations of every other particle. This keeps the particles apart, so lowers the correlation energy.” Thursday, November 24, 11 The quantitative explanation • introduce guiding center operators 1 @ 1 @ a¯i† = 2 z¯i⇤ ai† = 2 zi⇤ @z¯i @zi 1 @ 1 @ a¯i = 2 z¯i + ai = 2 zi + @z¯i⇤ @zi⇤ Guiding centers Landau orbits Conventional choice: z¯i = zi⇤ Thursday, November 24, 11 • The Laughlin state can be rewritten as q 1 z z¯ L = a¯† a¯† 0 = e 2 i i i j 0 i 1 L = (¯a† a¯†)(¯a a¯ )=0, 1, 2,... ij 2 i j i j No pair of particles has Lij Thursday, November 24, 11 z¯ does not have to be z⇤ ! • The guiding center geometry is fixed by the geometry of Coulomb equipotentials of a point charge, which do not have to be congruent to the shape of the Landau orbits • The second (but not the original) definition of the Laughlin state remains valid if z¯ ↵ z = ⇤ z¯⇤ ⇤ ↵⇤ z ✓ ◆ ✓ ◆✓ ◆ ↵⇤↵ ⇤ =1 (Bogoliubov SU(1,1) transformation) Thursday, November 24, 11 Physically z =constant z¯ =constant | | | | e- Shape of the “excluded region”of shape of the q flux quanta surrounding each Landau orbit electron The shape of the excluded region will self-select to minimize the correlation energy, but can fluctuate about that shape Thursday, November 24, 11 The quantum geometry The shape z¯ = constant is defined by a • | | UNIMODULAR 2D spatial metric g¯ab(r,t), detg ¯ =1 • Its Gaussian curvature is the curl of a DYNAMICAL spin- connection gauge field analogous to the STATIC spin- connection described by Wen and Zee (1992) in their treatment of FQHE on a static extrinsically-curved surface (like the sphere used in numerical diagonalization) • Here the surface is flat, the curved metric is NOT the induced metric from 3D Euclidean space, as in Wen and Zee Thursday, November 24, 11 Some key results • The filling factor p/q and the elementary fractional charge e* = e/q are joined by a third topological parameter, the GUIDING CENTER SPIN, s, which is quantized to integer or half-integer in an incompressible state by Gauss-Bonnet. • Charge density e s ⇢ (r)= B(r)+ ⇤ Kg(r) e H 2⇡ pe2 H = qh Gaussian curvature of gab(r) Thursday, November 24, 11 • leading term in Gaussian curvature is second derivative of metric. The zero-point fluctuations of the metric naturally reproduces the Q4 behavior of the structure function at long-wavelength found by Girvin MacDonald and Platzman. • The long-wavelength energy gap agrees quantitatively with GMP, in terms of the deformation stiffness of the state. Thursday, November 24, 11 • Core structures of quasi particles are calculable in terms of the shape-dependent correlation energy and the effective interaction between charge fluctuations (Gaussian curvature fluctuations). • The multicomponent case seems to have an independent metric field for each component Conclusion: the effective theory of FQHE is finally revealed as a marriage of Chern-Simons topology with (2D spatial) quantum geometry. Thursday, November 24, 11